3.17.38 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=448 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) \sqrt {d+e x}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)} \]

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Rubi [A]  time = 0.20, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} -\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) \sqrt {d+e x}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^(5/2)) + (2*(b*d - a*e
)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2)) - (10*b*(b*d
- a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) - (20*b^2*(b
*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (10*b^3*(
b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (2*b
^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (2*b^5*B*(d
+ e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x))

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^{7/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{7/2}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{5/2}}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^{3/2}}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 \sqrt {d+e x}}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) \sqrt {d+e x}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{3/2}}{e^6}+\frac {b^{10} B (d+e x)^{5/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {10 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 239, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-21 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+175 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-1050 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)-525 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)+35 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)-21 (b d-a e)^5 (B d-A e)+15 b^5 B (d+e x)^6\right )}{105 e^7 (a+b x) (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-21*(b*d - a*e)^5*(B*d - A*e) + 35*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x) -
525*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 1050*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e
)*(d + e*x)^3 + 175*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 21*b^4*(6*b*B*d - A*b*e - 5*a*B*
e)*(d + e*x)^5 + 15*b^5*B*(d + e*x)^6))/(105*e^7*(a + b*x)*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 33.34, size = 812, normalized size = 1.81 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (-21 b^5 B d^6+21 A b^5 e d^5+105 a b^4 B e d^5+210 b^5 B (d+e x) d^5-105 a A b^4 e^2 d^4-210 a^2 b^3 B e^2 d^4-1575 b^5 B (d+e x)^2 d^4-175 A b^5 e (d+e x) d^4-875 a b^4 B e (d+e x) d^4+210 a^2 A b^3 e^3 d^3+210 a^3 b^2 B e^3 d^3-2100 b^5 B (d+e x)^3 d^3+1050 A b^5 e (d+e x)^2 d^3+5250 a b^4 B e (d+e x)^2 d^3+700 a A b^4 e^2 (d+e x) d^3+1400 a^2 b^3 B e^2 (d+e x) d^3-210 a^3 A b^2 e^4 d^2-105 a^4 b B e^4 d^2+525 b^5 B (d+e x)^4 d^2+1050 A b^5 e (d+e x)^3 d^2+5250 a b^4 B e (d+e x)^3 d^2-3150 a A b^4 e^2 (d+e x)^2 d^2-6300 a^2 b^3 B e^2 (d+e x)^2 d^2-1050 a^2 A b^3 e^3 (d+e x) d^2-1050 a^3 b^2 B e^3 (d+e x) d^2+105 a^4 A b e^5 d+21 a^5 B e^5 d-126 b^5 B (d+e x)^5 d-175 A b^5 e (d+e x)^4 d-875 a b^4 B e (d+e x)^4 d-2100 a A b^4 e^2 (d+e x)^3 d-4200 a^2 b^3 B e^2 (d+e x)^3 d+3150 a^2 A b^3 e^3 (d+e x)^2 d+3150 a^3 b^2 B e^3 (d+e x)^2 d+700 a^3 A b^2 e^4 (d+e x) d+350 a^4 b B e^4 (d+e x) d-21 a^5 A e^6+15 b^5 B (d+e x)^6+21 A b^5 e (d+e x)^5+105 a b^4 B e (d+e x)^5+175 a A b^4 e^2 (d+e x)^4+350 a^2 b^3 B e^2 (d+e x)^4+1050 a^2 A b^3 e^3 (d+e x)^3+1050 a^3 b^2 B e^3 (d+e x)^3-1050 a^3 A b^2 e^4 (d+e x)^2-525 a^4 b B e^4 (d+e x)^2-175 a^4 A b e^5 (d+e x)-35 a^5 B e^5 (d+e x)\right )}{105 e^6 (d+e x)^{5/2} (a e+b x e)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(-21*b^5*B*d^6 + 21*A*b^5*d^5*e + 105*a*b^4*B*d^5*e - 105*a*A*b^4*d^4*e^2 - 210*a
^2*b^3*B*d^4*e^2 + 210*a^2*A*b^3*d^3*e^3 + 210*a^3*b^2*B*d^3*e^3 - 210*a^3*A*b^2*d^2*e^4 - 105*a^4*b*B*d^2*e^4
+ 105*a^4*A*b*d*e^5 + 21*a^5*B*d*e^5 - 21*a^5*A*e^6 + 210*b^5*B*d^5*(d + e*x) - 175*A*b^5*d^4*e*(d + e*x) - 8
75*a*b^4*B*d^4*e*(d + e*x) + 700*a*A*b^4*d^3*e^2*(d + e*x) + 1400*a^2*b^3*B*d^3*e^2*(d + e*x) - 1050*a^2*A*b^3
*d^2*e^3*(d + e*x) - 1050*a^3*b^2*B*d^2*e^3*(d + e*x) + 700*a^3*A*b^2*d*e^4*(d + e*x) + 350*a^4*b*B*d*e^4*(d +
e*x) - 175*a^4*A*b*e^5*(d + e*x) - 35*a^5*B*e^5*(d + e*x) - 1575*b^5*B*d^4*(d + e*x)^2 + 1050*A*b^5*d^3*e*(d
+ e*x)^2 + 5250*a*b^4*B*d^3*e*(d + e*x)^2 - 3150*a*A*b^4*d^2*e^2*(d + e*x)^2 - 6300*a^2*b^3*B*d^2*e^2*(d + e*x
)^2 + 3150*a^2*A*b^3*d*e^3*(d + e*x)^2 + 3150*a^3*b^2*B*d*e^3*(d + e*x)^2 - 1050*a^3*A*b^2*e^4*(d + e*x)^2 - 5
25*a^4*b*B*e^4*(d + e*x)^2 - 2100*b^5*B*d^3*(d + e*x)^3 + 1050*A*b^5*d^2*e*(d + e*x)^3 + 5250*a*b^4*B*d^2*e*(d
+ e*x)^3 - 2100*a*A*b^4*d*e^2*(d + e*x)^3 - 4200*a^2*b^3*B*d*e^2*(d + e*x)^3 + 1050*a^2*A*b^3*e^3*(d + e*x)^3
+ 1050*a^3*b^2*B*e^3*(d + e*x)^3 + 525*b^5*B*d^2*(d + e*x)^4 - 175*A*b^5*d*e*(d + e*x)^4 - 875*a*b^4*B*d*e*(d
+ e*x)^4 + 175*a*A*b^4*e^2*(d + e*x)^4 + 350*a^2*b^3*B*e^2*(d + e*x)^4 - 126*b^5*B*d*(d + e*x)^5 + 21*A*b^5*e
*(d + e*x)^5 + 105*a*b^4*B*e*(d + e*x)^5 + 15*b^5*B*(d + e*x)^6))/(105*e^6*(d + e*x)^(5/2)*(a*e + b*e*x))

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fricas [A]  time = 0.43, size = 592, normalized size = 1.32 \begin {gather*} \frac {2 \, {\left (15 \, B b^{5} e^{6} x^{6} - 3072 \, B b^{5} d^{6} - 21 \, A a^{5} e^{6} + 1792 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 4480 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 3360 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 280 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 14 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 3 \, {\left (12 \, B b^{5} d e^{5} - 7 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \, {\left (24 \, B b^{5} d^{2} e^{4} - 14 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 35 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 10 \, {\left (96 \, B b^{5} d^{3} e^{3} - 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 140 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} - 15 \, {\left (384 \, B b^{5} d^{4} e^{2} - 224 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 560 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 420 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 35 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 5 \, {\left (1536 \, B b^{5} d^{5} e - 896 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 2240 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 1680 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/105*(15*B*b^5*e^6*x^6 - 3072*B*b^5*d^6 - 21*A*a^5*e^6 + 1792*(5*B*a*b^4 + A*b^5)*d^5*e - 4480*(2*B*a^2*b^3 +
A*a*b^4)*d^4*e^2 + 3360*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 280*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - 14*(B*a^5 + 5
*A*a^4*b)*d*e^5 - 3*(12*B*b^5*d*e^5 - 7*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(24*B*b^5*d^2*e^4 - 14*(5*B*a*b^4 + A
*b^5)*d*e^5 + 35*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 10*(96*B*b^5*d^3*e^3 - 56*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 14
0*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - 105*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 - 15*(384*B*b^5*d^4*e^2 - 224*(5*B*a*b^
4 + A*b^5)*d^3*e^3 + 560*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 420*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 35*(B*a^4*b + 2
*A*a^3*b^2)*e^6)*x^2 - 5*(1536*B*b^5*d^5*e - 896*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 2240*(2*B*a^2*b^3 + A*a*b^4)*d^
3*e^3 - 1680*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 140*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + 7*(B*a^5 + 5*A*a^4*b)*e^6)*
x)*sqrt(e*x + d)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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giac [B]  time = 0.35, size = 1101, normalized size = 2.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*B*b^5*e^42*sgn(b*x + a) - 126*(x*e + d)^(5/2)*B*b^5*d*e^42*sgn(b*x + a) + 525*(x*e +
d)^(3/2)*B*b^5*d^2*e^42*sgn(b*x + a) - 2100*sqrt(x*e + d)*B*b^5*d^3*e^42*sgn(b*x + a) + 105*(x*e + d)^(5/2)*B
*a*b^4*e^43*sgn(b*x + a) + 21*(x*e + d)^(5/2)*A*b^5*e^43*sgn(b*x + a) - 875*(x*e + d)^(3/2)*B*a*b^4*d*e^43*sgn
(b*x + a) - 175*(x*e + d)^(3/2)*A*b^5*d*e^43*sgn(b*x + a) + 5250*sqrt(x*e + d)*B*a*b^4*d^2*e^43*sgn(b*x + a) +
1050*sqrt(x*e + d)*A*b^5*d^2*e^43*sgn(b*x + a) + 350*(x*e + d)^(3/2)*B*a^2*b^3*e^44*sgn(b*x + a) + 175*(x*e +
d)^(3/2)*A*a*b^4*e^44*sgn(b*x + a) - 4200*sqrt(x*e + d)*B*a^2*b^3*d*e^44*sgn(b*x + a) - 2100*sqrt(x*e + d)*A*
a*b^4*d*e^44*sgn(b*x + a) + 1050*sqrt(x*e + d)*B*a^3*b^2*e^45*sgn(b*x + a) + 1050*sqrt(x*e + d)*A*a^2*b^3*e^45
*sgn(b*x + a))*e^(-49) - 2/15*(225*(x*e + d)^2*B*b^5*d^4*sgn(b*x + a) - 30*(x*e + d)*B*b^5*d^5*sgn(b*x + a) +
3*B*b^5*d^6*sgn(b*x + a) - 750*(x*e + d)^2*B*a*b^4*d^3*e*sgn(b*x + a) - 150*(x*e + d)^2*A*b^5*d^3*e*sgn(b*x +
a) + 125*(x*e + d)*B*a*b^4*d^4*e*sgn(b*x + a) + 25*(x*e + d)*A*b^5*d^4*e*sgn(b*x + a) - 15*B*a*b^4*d^5*e*sgn(b
*x + a) - 3*A*b^5*d^5*e*sgn(b*x + a) + 900*(x*e + d)^2*B*a^2*b^3*d^2*e^2*sgn(b*x + a) + 450*(x*e + d)^2*A*a*b^
4*d^2*e^2*sgn(b*x + a) - 200*(x*e + d)*B*a^2*b^3*d^3*e^2*sgn(b*x + a) - 100*(x*e + d)*A*a*b^4*d^3*e^2*sgn(b*x
+ a) + 30*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 15*A*a*b^4*d^4*e^2*sgn(b*x + a) - 450*(x*e + d)^2*B*a^3*b^2*d*e^3*s
gn(b*x + a) - 450*(x*e + d)^2*A*a^2*b^3*d*e^3*sgn(b*x + a) + 150*(x*e + d)*B*a^3*b^2*d^2*e^3*sgn(b*x + a) + 15
0*(x*e + d)*A*a^2*b^3*d^2*e^3*sgn(b*x + a) - 30*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 30*A*a^2*b^3*d^3*e^3*sgn(b*x
+ a) + 75*(x*e + d)^2*B*a^4*b*e^4*sgn(b*x + a) + 150*(x*e + d)^2*A*a^3*b^2*e^4*sgn(b*x + a) - 50*(x*e + d)*B*a
^4*b*d*e^4*sgn(b*x + a) - 100*(x*e + d)*A*a^3*b^2*d*e^4*sgn(b*x + a) + 15*B*a^4*b*d^2*e^4*sgn(b*x + a) + 30*A*
a^3*b^2*d^2*e^4*sgn(b*x + a) + 5*(x*e + d)*B*a^5*e^5*sgn(b*x + a) + 25*(x*e + d)*A*a^4*b*e^5*sgn(b*x + a) - 3*
B*a^5*d*e^5*sgn(b*x + a) - 15*A*a^4*b*d*e^5*sgn(b*x + a) + 3*A*a^5*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^(5/2)

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maple [A]  time = 0.06, size = 689, normalized size = 1.54 \begin {gather*} -\frac {2 \left (-15 B \,b^{5} e^{6} x^{6}-21 A \,b^{5} e^{6} x^{5}-105 B a \,b^{4} e^{6} x^{5}+36 B \,b^{5} d \,e^{5} x^{5}-175 A a \,b^{4} e^{6} x^{4}+70 A \,b^{5} d \,e^{5} x^{4}-350 B \,a^{2} b^{3} e^{6} x^{4}+350 B a \,b^{4} d \,e^{5} x^{4}-120 B \,b^{5} d^{2} e^{4} x^{4}-1050 A \,a^{2} b^{3} e^{6} x^{3}+1400 A a \,b^{4} d \,e^{5} x^{3}-560 A \,b^{5} d^{2} e^{4} x^{3}-1050 B \,a^{3} b^{2} e^{6} x^{3}+2800 B \,a^{2} b^{3} d \,e^{5} x^{3}-2800 B a \,b^{4} d^{2} e^{4} x^{3}+960 B \,b^{5} d^{3} e^{3} x^{3}+1050 A \,a^{3} b^{2} e^{6} x^{2}-6300 A \,a^{2} b^{3} d \,e^{5} x^{2}+8400 A a \,b^{4} d^{2} e^{4} x^{2}-3360 A \,b^{5} d^{3} e^{3} x^{2}+525 B \,a^{4} b \,e^{6} x^{2}-6300 B \,a^{3} b^{2} d \,e^{5} x^{2}+16800 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-16800 B a \,b^{4} d^{3} e^{3} x^{2}+5760 B \,b^{5} d^{4} e^{2} x^{2}+175 A \,a^{4} b \,e^{6} x +1400 A \,a^{3} b^{2} d \,e^{5} x -8400 A \,a^{2} b^{3} d^{2} e^{4} x +11200 A a \,b^{4} d^{3} e^{3} x -4480 A \,b^{5} d^{4} e^{2} x +35 B \,a^{5} e^{6} x +700 B \,a^{4} b d \,e^{5} x -8400 B \,a^{3} b^{2} d^{2} e^{4} x +22400 B \,a^{2} b^{3} d^{3} e^{3} x -22400 B a \,b^{4} d^{4} e^{2} x +7680 B \,b^{5} d^{5} e x +21 A \,a^{5} e^{6}+70 A \,a^{4} b d \,e^{5}+560 A \,a^{3} b^{2} d^{2} e^{4}-3360 A \,a^{2} b^{3} d^{3} e^{3}+4480 A a \,b^{4} d^{4} e^{2}-1792 A \,b^{5} d^{5} e +14 B \,a^{5} d \,e^{5}+280 B \,a^{4} b \,d^{2} e^{4}-3360 B \,a^{3} b^{2} d^{3} e^{3}+8960 B \,a^{2} b^{3} d^{4} e^{2}-8960 B a \,b^{4} d^{5} e +3072 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{105 \left (e x +d \right )^{\frac {5}{2}} \left (b x +a \right )^{5} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x)

[Out]

-2/105/(e*x+d)^(5/2)*(-15*B*b^5*e^6*x^6-21*A*b^5*e^6*x^5-105*B*a*b^4*e^6*x^5+36*B*b^5*d*e^5*x^5-175*A*a*b^4*e^
6*x^4+70*A*b^5*d*e^5*x^4-350*B*a^2*b^3*e^6*x^4+350*B*a*b^4*d*e^5*x^4-120*B*b^5*d^2*e^4*x^4-1050*A*a^2*b^3*e^6*
x^3+1400*A*a*b^4*d*e^5*x^3-560*A*b^5*d^2*e^4*x^3-1050*B*a^3*b^2*e^6*x^3+2800*B*a^2*b^3*d*e^5*x^3-2800*B*a*b^4*
d^2*e^4*x^3+960*B*b^5*d^3*e^3*x^3+1050*A*a^3*b^2*e^6*x^2-6300*A*a^2*b^3*d*e^5*x^2+8400*A*a*b^4*d^2*e^4*x^2-336
0*A*b^5*d^3*e^3*x^2+525*B*a^4*b*e^6*x^2-6300*B*a^3*b^2*d*e^5*x^2+16800*B*a^2*b^3*d^2*e^4*x^2-16800*B*a*b^4*d^3
*e^3*x^2+5760*B*b^5*d^4*e^2*x^2+175*A*a^4*b*e^6*x+1400*A*a^3*b^2*d*e^5*x-8400*A*a^2*b^3*d^2*e^4*x+11200*A*a*b^
4*d^3*e^3*x-4480*A*b^5*d^4*e^2*x+35*B*a^5*e^6*x+700*B*a^4*b*d*e^5*x-8400*B*a^3*b^2*d^2*e^4*x+22400*B*a^2*b^3*d
^3*e^3*x-22400*B*a*b^4*d^4*e^2*x+7680*B*b^5*d^5*e*x+21*A*a^5*e^6+70*A*a^4*b*d*e^5+560*A*a^3*b^2*d^2*e^4-3360*A
*a^2*b^3*d^3*e^3+4480*A*a*b^4*d^4*e^2-1792*A*b^5*d^5*e+14*B*a^5*d*e^5+280*B*a^4*b*d^2*e^4-3360*B*a^3*b^2*d^3*e
^3+8960*B*a^2*b^3*d^4*e^2-8960*B*a*b^4*d^5*e+3072*B*b^5*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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maxima [A]  time = 0.78, size = 647, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \, {\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} A}{15 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (15 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 8960 \, a b^{4} d^{5} e - 8960 \, a^{2} b^{3} d^{4} e^{2} + 3360 \, a^{3} b^{2} d^{3} e^{3} - 280 \, a^{4} b d^{2} e^{4} - 14 \, a^{5} d e^{5} - 3 \, {\left (12 \, b^{5} d e^{5} - 35 \, a b^{4} e^{6}\right )} x^{5} + 10 \, {\left (12 \, b^{5} d^{2} e^{4} - 35 \, a b^{4} d e^{5} + 35 \, a^{2} b^{3} e^{6}\right )} x^{4} - 10 \, {\left (96 \, b^{5} d^{3} e^{3} - 280 \, a b^{4} d^{2} e^{4} + 280 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} - 15 \, {\left (384 \, b^{5} d^{4} e^{2} - 1120 \, a b^{4} d^{3} e^{3} + 1120 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 35 \, a^{4} b e^{6}\right )} x^{2} - 5 \, {\left (1536 \, b^{5} d^{5} e - 4480 \, a b^{4} d^{4} e^{2} + 4480 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 140 \, a^{4} b d e^{5} + 7 \, a^{5} e^{6}\right )} x\right )} B}{105 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt {e x + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/15*(3*b^5*e^5*x^5 + 256*b^5*d^5 - 640*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 - 80*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 3*a^5*e^5 - 5*(2*b^5*d*e^4 - 5*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 20*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 +
30*(16*b^5*d^3*e^2 - 40*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 - 5*a^3*b^2*e^5)*x^2 + 5*(128*b^5*d^4*e - 320*a*b^4*
d^3*e^2 + 240*a^2*b^3*d^2*e^3 - 40*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*A/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(e*x
+ d)) + 2/105*(15*b^5*e^6*x^6 - 3072*b^5*d^6 + 8960*a*b^4*d^5*e - 8960*a^2*b^3*d^4*e^2 + 3360*a^3*b^2*d^3*e^3
- 280*a^4*b*d^2*e^4 - 14*a^5*d*e^5 - 3*(12*b^5*d*e^5 - 35*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 35*a*b^4*d*e^
5 + 35*a^2*b^3*e^6)*x^4 - 10*(96*b^5*d^3*e^3 - 280*a*b^4*d^2*e^4 + 280*a^2*b^3*d*e^5 - 105*a^3*b^2*e^6)*x^3 -
15*(384*b^5*d^4*e^2 - 1120*a*b^4*d^3*e^3 + 1120*a^2*b^3*d^2*e^4 - 420*a^3*b^2*d*e^5 + 35*a^4*b*e^6)*x^2 - 5*(1
536*b^5*d^5*e - 4480*a*b^4*d^4*e^2 + 4480*a^2*b^3*d^3*e^3 - 1680*a^3*b^2*d^2*e^4 + 140*a^4*b*d*e^5 + 7*a^5*e^6
)*x)*B/((e^9*x^2 + 2*d*e^8*x + d^2*e^7)*sqrt(e*x + d))

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mupad [B]  time = 3.95, size = 718, normalized size = 1.60 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {x^3\,\left (20\,B\,a^3\,b^2\,e^6-\frac {160\,B\,a^2\,b^3\,d\,e^5}{3}+20\,A\,a^2\,b^3\,e^6+\frac {160\,B\,a\,b^4\,d^2\,e^4}{3}-\frac {80\,A\,a\,b^4\,d\,e^5}{3}-\frac {128\,B\,b^5\,d^3\,e^3}{7}+\frac {32\,A\,b^5\,d^2\,e^4}{3}\right )}{b\,e^9}-\frac {x^2\,\left (10\,B\,a^4\,b\,e^6-120\,B\,a^3\,b^2\,d\,e^5+20\,A\,a^3\,b^2\,e^6+320\,B\,a^2\,b^3\,d^2\,e^4-120\,A\,a^2\,b^3\,d\,e^5-320\,B\,a\,b^4\,d^3\,e^3+160\,A\,a\,b^4\,d^2\,e^4+\frac {768\,B\,b^5\,d^4\,e^2}{7}-64\,A\,b^5\,d^3\,e^3\right )}{b\,e^9}-\frac {\frac {4\,B\,a^5\,d\,e^5}{15}+\frac {2\,A\,a^5\,e^6}{5}+\frac {16\,B\,a^4\,b\,d^2\,e^4}{3}+\frac {4\,A\,a^4\,b\,d\,e^5}{3}-64\,B\,a^3\,b^2\,d^3\,e^3+\frac {32\,A\,a^3\,b^2\,d^2\,e^4}{3}+\frac {512\,B\,a^2\,b^3\,d^4\,e^2}{3}-64\,A\,a^2\,b^3\,d^3\,e^3-\frac {512\,B\,a\,b^4\,d^5\,e}{3}+\frac {256\,A\,a\,b^4\,d^4\,e^2}{3}+\frac {2048\,B\,b^5\,d^6}{35}-\frac {512\,A\,b^5\,d^5\,e}{15}}{b\,e^9}+\frac {b^3\,x^5\,\left (\frac {2\,A\,b\,e}{5}+2\,B\,a\,e-\frac {24\,B\,b\,d}{35}\right )}{e^4}-\frac {x\,\left (70\,B\,a^5\,e^6+1400\,B\,a^4\,b\,d\,e^5+350\,A\,a^4\,b\,e^6-16800\,B\,a^3\,b^2\,d^2\,e^4+2800\,A\,a^3\,b^2\,d\,e^5+44800\,B\,a^2\,b^3\,d^3\,e^3-16800\,A\,a^2\,b^3\,d^2\,e^4-44800\,B\,a\,b^4\,d^4\,e^2+22400\,A\,a\,b^4\,d^3\,e^3+15360\,B\,b^5\,d^5\,e-8960\,A\,b^5\,d^4\,e^2\right )}{105\,b\,e^9}+\frac {b^2\,x^4\,\left (\frac {20\,B\,a^2\,e^2}{3}-\frac {20\,B\,a\,b\,d\,e}{3}+\frac {10\,A\,a\,b\,e^2}{3}+\frac {16\,B\,b^2\,d^2}{7}-\frac {4\,A\,b^2\,d\,e}{3}\right )}{e^5}+\frac {2\,B\,b^4\,x^6}{7\,e^3}\right )}{x^3\,\sqrt {d+e\,x}+\frac {a\,d^2\,\sqrt {d+e\,x}}{b\,e^2}+\frac {x^2\,\left (a\,e^9+2\,b\,d\,e^8\right )\,\sqrt {d+e\,x}}{b\,e^9}+\frac {d\,x\,\left (2\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^(7/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((x^3*(20*A*a^2*b^3*e^6 + 20*B*a^3*b^2*e^6 + (32*A*b^5*d^2*e^4)/3 - (128*B*b^
5*d^3*e^3)/7 + (160*B*a*b^4*d^2*e^4)/3 - (160*B*a^2*b^3*d*e^5)/3 - (80*A*a*b^4*d*e^5)/3))/(b*e^9) - (x^2*(10*B
*a^4*b*e^6 + 20*A*a^3*b^2*e^6 - 64*A*b^5*d^3*e^3 + (768*B*b^5*d^4*e^2)/7 + 160*A*a*b^4*d^2*e^4 - 120*A*a^2*b^3
*d*e^5 - 320*B*a*b^4*d^3*e^3 - 120*B*a^3*b^2*d*e^5 + 320*B*a^2*b^3*d^2*e^4))/(b*e^9) - ((2*A*a^5*e^6)/5 + (204
8*B*b^5*d^6)/35 - (512*A*b^5*d^5*e)/15 + (4*B*a^5*d*e^5)/15 + (256*A*a*b^4*d^4*e^2)/3 + (16*B*a^4*b*d^2*e^4)/3
- 64*A*a^2*b^3*d^3*e^3 + (32*A*a^3*b^2*d^2*e^4)/3 + (512*B*a^2*b^3*d^4*e^2)/3 - 64*B*a^3*b^2*d^3*e^3 + (4*A*a
^4*b*d*e^5)/3 - (512*B*a*b^4*d^5*e)/3)/(b*e^9) + (b^3*x^5*((2*A*b*e)/5 + 2*B*a*e - (24*B*b*d)/35))/e^4 - (x*(7
0*B*a^5*e^6 + 350*A*a^4*b*e^6 + 15360*B*b^5*d^5*e - 8960*A*b^5*d^4*e^2 + 22400*A*a*b^4*d^3*e^3 + 2800*A*a^3*b^
2*d*e^5 - 44800*B*a*b^4*d^4*e^2 - 16800*A*a^2*b^3*d^2*e^4 + 44800*B*a^2*b^3*d^3*e^3 - 16800*B*a^3*b^2*d^2*e^4
+ 1400*B*a^4*b*d*e^5))/(105*b*e^9) + (b^2*x^4*((20*B*a^2*e^2)/3 + (16*B*b^2*d^2)/7 + (10*A*a*b*e^2)/3 - (4*A*b
^2*d*e)/3 - (20*B*a*b*d*e)/3))/e^5 + (2*B*b^4*x^6)/(7*e^3)))/(x^3*(d + e*x)^(1/2) + (a*d^2*(d + e*x)^(1/2))/(b
*e^2) + (x^2*(a*e^9 + 2*b*d*e^8)*(d + e*x)^(1/2))/(b*e^9) + (d*x*(2*a*e + b*d)*(d + e*x)^(1/2))/(b*e^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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